1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
|
# polarization grating from C. Oh and M.J. Escuti, Optics Letters, Vol. 33, No. 20, pp. 2287-9, 2008
# note: reference uses z as the propagation direction and y as the out-of-plane direction; this script uses x and z, respectively
import math
import matplotlib.pyplot as plt
import numpy as np
import meep as mp
resolution = 50 # pixels/μm
dpml = 1.0 # PML thickness
dsub = 1.0 # substrate thickness
dpad = 1.0 # padding thickness
k_point = mp.Vector3(0, 0, 0)
pml_layers = [mp.PML(thickness=dpml, direction=mp.X)]
n_0 = 1.55
delta_n = 0.159
epsilon_diag = mp.Matrix(
mp.Vector3(n_0**2, 0, 0),
mp.Vector3(0, n_0**2, 0),
mp.Vector3(0, 0, (n_0 + delta_n) ** 2),
)
wvl = 0.54 # center wavelength
fcen = 1 / wvl # center frequency
def pol_grating(d, ph, gp, nmode):
sx = dpml + dsub + d + d + dpad + dpml
sy = gp
cell_size = mp.Vector3(sx, sy, 0)
# twist angle of nematic director; from equation 1b
def phi(p):
xx = p.x - (-0.5 * sx + dpml + dsub)
if (xx >= 0) and (xx <= d):
return math.pi * p.y / gp + ph * xx / d
else:
return math.pi * p.y / gp - ph * xx / d + 2 * ph
# return the anisotropic permittivity tensor for a uniaxial, twisted nematic liquid crystal
def lc_mat(p):
# rotation matrix for rotation around x axis
Rx = mp.Matrix(
mp.Vector3(1, 0, 0),
mp.Vector3(0, math.cos(phi(p)), math.sin(phi(p))),
mp.Vector3(0, -math.sin(phi(p)), math.cos(phi(p))),
)
lc_epsilon = Rx * epsilon_diag * Rx.transpose()
lc_epsilon_diag = mp.Vector3(lc_epsilon[0].x, lc_epsilon[1].y, lc_epsilon[2].z)
lc_epsilon_offdiag = mp.Vector3(
lc_epsilon[1].x, lc_epsilon[2].x, lc_epsilon[2].y
)
return mp.Medium(
epsilon_diag=lc_epsilon_diag, epsilon_offdiag=lc_epsilon_offdiag
)
geometry = [
mp.Block(
center=mp.Vector3(-0.5 * sx + 0.5 * (dpml + dsub)),
size=mp.Vector3(dpml + dsub, mp.inf, mp.inf),
material=mp.Medium(index=n_0),
),
mp.Block(
center=mp.Vector3(-0.5 * sx + dpml + dsub + d),
size=mp.Vector3(2 * d, mp.inf, mp.inf),
material=lc_mat,
),
]
# linear-polarized planewave pulse source
src_pt = mp.Vector3(-0.5 * sx + dpml + 0.3 * dsub, 0, 0)
sources = [
mp.Source(
mp.GaussianSource(fcen, fwidth=0.05 * fcen),
component=mp.Ez,
center=src_pt,
size=mp.Vector3(0, sy, 0),
),
mp.Source(
mp.GaussianSource(fcen, fwidth=0.05 * fcen),
component=mp.Ey,
center=src_pt,
size=mp.Vector3(0, sy, 0),
),
]
sim = mp.Simulation(
resolution=resolution,
cell_size=cell_size,
boundary_layers=pml_layers,
k_point=k_point,
sources=sources,
default_material=mp.Medium(index=n_0),
)
tran_pt = mp.Vector3(0.5 * sx - dpml - 0.5 * dpad, 0, 0)
tran_flux = sim.add_flux(
fcen, 0, 1, mp.FluxRegion(center=tran_pt, size=mp.Vector3(0, sy, 0))
)
sim.run(until_after_sources=100)
input_flux = mp.get_fluxes(tran_flux)
sim.reset_meep()
sim = mp.Simulation(
resolution=resolution,
cell_size=cell_size,
boundary_layers=pml_layers,
k_point=k_point,
sources=sources,
geometry=geometry,
)
tran_flux = sim.add_flux(
fcen, 0, 1, mp.FluxRegion(center=tran_pt, size=mp.Vector3(0, sy, 0))
)
sim.run(until_after_sources=300)
res1 = sim.get_eigenmode_coefficients(
tran_flux, range(1, nmode + 1), eig_parity=mp.ODD_Z + mp.EVEN_Y
)
res2 = sim.get_eigenmode_coefficients(
tran_flux, range(1, nmode + 1), eig_parity=mp.EVEN_Z + mp.ODD_Y
)
angles = [math.degrees(math.acos(kdom.x / fcen)) for kdom in res1.kdom]
return input_flux[0], angles, res1.alpha[:, 0, 0], res2.alpha[:, 0, 0]
ph_uniaxial = 0 # chiral layer twist angle for uniaxial grating
ph_twisted = 70 # chiral layer twist angle for bilayer grating
gp = 6.5 # grating period
nmode = 5 # number of mode coefficients to compute
dd = np.arange(0.1, 3.5, 0.1) # chiral layer thickness
m0_uniaxial = np.zeros(dd.size)
m1_uniaxial = np.zeros(dd.size)
ang_uniaxial = np.zeros(dd.size)
m0_twisted = np.zeros(dd.size)
m1_twisted = np.zeros(dd.size)
ang_twisted = np.zeros(dd.size)
for k in range(len(dd)):
input_flux, angles, coeffs1, coeffs2 = pol_grating(
0.5 * dd[k], math.radians(ph_uniaxial), gp, nmode
)
tran = (abs(coeffs1) ** 2 + abs(coeffs2) ** 2) / input_flux
for m in range(nmode):
print(f"tran (uniaxial):, {m}, {angles[m]:.2f}, {tran[m]:.5f}")
m0_uniaxial[k] = tran[0]
m1_uniaxial[k] = tran[1]
ang_uniaxial[k] = angles[1]
input_flux, angles, coeffs1, coeffs2 = pol_grating(
dd[k], math.radians(ph_twisted), gp, nmode
)
tran = (abs(coeffs1) ** 2 + abs(coeffs2) ** 2) / input_flux
for m in range(nmode):
print(f"tran (twisted):, {m}, {angles[m]:.2f}, {tran[m]:.5f}")
m0_twisted[k] = tran[0]
m1_twisted[k] = tran[1]
ang_twisted[k] = angles[1]
cos_angles = [math.cos(math.radians(t)) for t in ang_uniaxial]
tran = m0_uniaxial + 2 * m1_uniaxial
eff_m0 = m0_uniaxial / tran
eff_m1 = (2 * m1_uniaxial / tran) / cos_angles
phase = delta_n * dd / wvl
eff_m0_analytic = [math.cos(math.pi * p) ** 2 for p in phase]
eff_m1_analytic = [math.sin(math.pi * p) ** 2 for p in phase]
plt.figure(dpi=150)
plt.subplot(1, 2, 1)
plt.plot(phase, eff_m0, "bo-", clip_on=False, label="0th order (meep)")
plt.plot(phase, eff_m0_analytic, "b--", clip_on=False, label="0th order (analytic)")
plt.plot(phase, eff_m1, "ro-", clip_on=False, label="±1 orders (meep)")
plt.plot(phase, eff_m1_analytic, "r--", clip_on=False, label="±1 orders (analytic)")
plt.axis([0, 1.0, 0, 1])
plt.xticks(list(np.arange(0, 1.2, 0.2)))
plt.xlabel("phase delay Δnd/λ")
plt.ylabel("diffraction efficiency @ λ = 0.54 μm")
plt.legend(loc="center")
plt.title("homogeneous uniaxial grating")
cos_angles = [math.cos(math.radians(t)) for t in ang_twisted]
tran = m0_twisted + 2 * m1_twisted
eff_m0 = m0_twisted / tran
eff_m1 = (2 * m1_twisted / tran) / cos_angles
plt.subplot(1, 2, 2)
plt.plot(phase, eff_m0, "bo-", clip_on=False, label="0th order (meep)")
plt.plot(phase, eff_m1, "ro-", clip_on=False, label="±1 orders (meep)")
plt.axis([0, 1.0, 0, 1])
plt.xticks(list(np.arange(0, 1.2, 0.2)))
plt.xlabel("phase delay Δnd/λ")
plt.ylabel("diffraction efficiency @ λ = 0.54 μm")
plt.legend(loc="center")
plt.title("bilayer twisted-nematic grating")
plt.show()
|
